Theorem mulnqprlemru 7477

Description: Lemma for mulnqpr 7480. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)

Assertion

Ref	Expression
mulnqprlemru	|- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )

Distinct variable groups:   A, l, u   B, l, u

Proof of Theorem mulnqprlemru

Dummy variables f g h r s t x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nqprlu 7450	. . . . . 6 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
2		nqprlu 7450	. . . . . 6 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
3		df-imp 7372	. . . . . . 7 |- .P. = ( x e. P. , y e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` x ) /\ h e. ( 1st ` y ) /\ f = ( g .Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` x ) /\ h e. ( 2nd ` y ) /\ f = ( g .Q h ) ) } >. )
4		mulclnq 7279	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
5	3, 4	genpelvu 7416	. . . . . 6 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. e. P. /\ <. { l | l <Q B } , { u | B <Q u } >. e. P. ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) <-> E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) ) )
6	1, 2, 5	syl2an 287	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) <-> E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) ) )
7	6	biimpa 294	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) )
8		vex 2715	. . . . . . . . . . . . 13 |- s e. _V
9		breq2 3969	. . . . . . . . . . . . 13 |- ( u = s -> ( A <Q u <-> A <Q s ) )
10		ltnqex 7452	. . . . . . . . . . . . . 14 |- { l | l <Q A } e. _V
11		gtnqex 7453	. . . . . . . . . . . . . 14 |- { u | A <Q u } e. _V
12	10, 11	op2nd 6089	. . . . . . . . . . . . 13 |- ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) = { u | A <Q u }
13	8, 9, 12	elab2 2860	. . . . . . . . . . . 12 |- ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> A <Q s )
14	13	biimpi 119	. . . . . . . . . . 11 |- ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) -> A <Q s )
15	14	ad2antrl 482	. . . . . . . . . 10 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> A <Q s )
16	15	adantr 274	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> A <Q s )
17		vex 2715	. . . . . . . . . . . . 13 |- t e. _V
18		breq2 3969	. . . . . . . . . . . . 13 |- ( u = t -> ( B <Q u <-> B <Q t ) )
19		ltnqex 7452	. . . . . . . . . . . . . 14 |- { l | l <Q B } e. _V
20		gtnqex 7453	. . . . . . . . . . . . . 14 |- { u | B <Q u } e. _V
21	19, 20	op2nd 6089	. . . . . . . . . . . . 13 |- ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) = { u | B <Q u }
22	17, 18, 21	elab2 2860	. . . . . . . . . . . 12 |- ( t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) <-> B <Q t )
23	22	biimpi 119	. . . . . . . . . . 11 |- ( t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) -> B <Q t )
24	23	ad2antll 483	. . . . . . . . . 10 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> B <Q t )
25	24	adantr 274	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> B <Q t )
26		ltrelnq 7268	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
27	26	brel 4635	. . . . . . . . . . 11 |- ( A <Q s -> ( A e. Q. /\ s e. Q. ) )
28	16, 27	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( A e. Q. /\ s e. Q. ) )
29	26	brel 4635	. . . . . . . . . . 11 |- ( B <Q t -> ( B e. Q. /\ t e. Q. ) )
30	25, 29	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( B e. Q. /\ t e. Q. ) )
31		lt2mulnq 7308	. . . . . . . . . 10 |- ( ( ( A e. Q. /\ s e. Q. ) /\ ( B e. Q. /\ t e. Q. ) ) -> ( ( A <Q s /\ B <Q t ) -> ( A .Q B ) <Q ( s .Q t ) ) )
32	28, 30, 31	syl2anc 409	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( ( A <Q s /\ B <Q t ) -> ( A .Q B ) <Q ( s .Q t ) ) )
33	16, 25, 32	mp2and 430	. . . . . . . 8 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( A .Q B ) <Q ( s .Q t ) )
34		breq2 3969	. . . . . . . . 9 |- ( r = ( s .Q t ) -> ( ( A .Q B ) <Q r <-> ( A .Q B ) <Q ( s .Q t ) ) )
35	34	adantl 275	. . . . . . . 8 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( ( A .Q B ) <Q r <-> ( A .Q B ) <Q ( s .Q t ) ) )
36	33, 35	mpbird 166	. . . . . . 7 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( A .Q B ) <Q r )
37		vex 2715	. . . . . . . 8 |- r e. _V
38		breq2 3969	. . . . . . . 8 |- ( u = r -> ( ( A .Q B ) <Q u <-> ( A .Q B ) <Q r ) )
39		ltnqex 7452	. . . . . . . . 9 |- { l | l <Q ( A .Q B ) } e. _V
40		gtnqex 7453	. . . . . . . . 9 |- { u | ( A .Q B ) <Q u } e. _V
41	39, 40	op2nd 6089	. . . . . . . 8 |- ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) = { u | ( A .Q B ) <Q u }
42	37, 38, 41	elab2 2860	. . . . . . 7 |- ( r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) <-> ( A .Q B ) <Q r )
43	36, 42	sylibr 133	. . . . . 6 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )
44	43	ex 114	. . . . 5 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> ( r = ( s .Q t ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) )
45	44	rexlimdvva 2582	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> ( E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) )
46	7, 45	mpd 13	. . 3 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )
47	46	ex 114	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) )
48	47	ssrdv 3134	1 |- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   {cab 2143   E.wrex 2436   C_ wss 3102   <.cop 3563   class class class wbr 3965   ` cfv 5167  (class class class)co 5818   2ndc2nd 6081   Q.cnq 7183   .Q cmq 7186   <Q cltq 7188   P.cnp 7194   .P. cmp 7197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4248  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-irdg 6311  df-1o 6357  df-oadd 6361  df-omul 6362  df-er 6473  df-ec 6475  df-qs 6479  df-ni 7207  df-pli 7208  df-mi 7209  df-lti 7210  df-plpq 7247  df-mpq 7248  df-enq 7250  df-nqqs 7251  df-plqqs 7252  df-mqqs 7253  df-1nqqs 7254  df-rq 7255  df-ltnqqs 7256  df-inp 7369  df-imp 7372

This theorem is referenced by:  mulnqprlemfl  7478  mulnqpr  7480